Complete combustion happens when there is enough oxygen that will be in charge of using up all of the available reactants. Incomplete combustion will take place when there is a lack of oxygen, so what happens is that there will be smoke that will be produced in the process.
Take note that in terms of energy, there will be more energy that will be available with complete combustion as compared to incomplete combustion. One of the things that you will notice is if it is incomplete combustion, there are still other byproducts that will be left behind. In complete combustion, you can expect that nothing will be left behind.